Nonlinear growth (and breakdown) of disturbances in developing Hagen Poiseuille flow

Abstract

The effect of imposing disturbances on developing Hagen Poiseuille flow is investigated for large Reynolds numbers. A formal large Reynolds number analysis is used throughout, which leads to a parabolic approximation (to the Navier-Stokes equations). In this inlet region [which is long O(Reynoldsnumber)×pipe radius], assuming uniform/near uniform inlet flow conditions, the boundary layers develop on the walls of the pipe, that eventually merge. Boundary layers are known to be susceptible to three-dimensional eigensolutions that grow algebraically in the streamwise direction. In this study, nonaxisymmetric disturbances are triggered through (i) the imposition of the aforementioned eigenstates at the pipe inlet and (ii) forcing the azimuthal velocity on the pipe wall. Fully nonlinear, steady disturbances are considered in detail; if the disturbance amplitude is sufficiently large, a solution “breakdown” is observed (associated with a rapid growth of the radial and azimuthal velocity components close to lines of symmetry). This appears to be linked to reversals in both the azimuthal and radial velocity components, suggesting a possible mechanism for flow transition. Some comparison is also made with the analogous effect on planar (Blasius-type) boundary layers. The analysis is wholly rational, with (in particular) the highly nonlinear and nonparallel flow effects being treated in an entirely consistent manner.